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Theorem psubclsubN 35744
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s 𝑆 = (PSubSp‘𝐾)
psubclsub.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclsubN ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2771 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
2 psubclsub.c . . 3 𝐶 = (PSubCl‘𝐾)
31, 2psubcli2N 35743 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)
4 eqid 2771 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
54, 1, 2psubcliN 35742 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
65simpld 476 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
84, 7, 1polsubN 35711 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
96, 8syldan 571 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
104, 7psubssat 35558 . . . 4 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
119, 10syldan 571 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
124, 7, 1polsubN 35711 . . 3 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
1311, 12syldan 571 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
143, 13eqeltrrd 2851 1 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wss 3723  cfv 6031  Atomscatm 35068  HLchlt 35155  PSubSpcpsubsp 35300  𝑃cpolN 35706  PSubClcpscN 35738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095  ax-riotaBAD 34757
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-undef 7550  df-preset 17135  df-poset 17153  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34981  df-ol 34983  df-oml 34984  df-ats 35072  df-atl 35103  df-cvlat 35127  df-hlat 35156  df-psubsp 35307  df-pmap 35308  df-polarityN 35707  df-psubclN 35739
This theorem is referenced by:  pclfinclN  35754
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